Problem: Determine the value of the expression
\[\log_2 (27 + \log_2 (27 + \log_2 (27 + \cdots))),\]assuming it is positive.
Explanation: Let
\[x = \log_2 (27 + \log_2 (27 + \log_2 (27 + \dotsb))).\]Then
\[x = \log_2 (27 + x),\]so $2^x = x + 27.$

To solve this equation, we plot $y = 2^x$ and $y = x + 27.$

[asy]
unitsize(0.15 cm);

real func (real x) {
  return(2^x);
}

draw(graph(func,-30,log(40)/log(2)),red);
draw((-30,-3)--(13,40),blue);
draw((-30,0)--(13,0));
draw((0,-5)--(0,40));

dot("$(5,32)$", (5,32), SE);
label("$y = 2^x$", (10,16));
label("$y = x + 27$", (-18,18));
[/asy]

By inspection, the graphs intersect at $(5,32).$  Beyond this point, the graph of $y = 2^x$ increases much faster than the graph of $y = x + 27,$ so the only positive solution is $x = \boxed{5}.$